Skew Hadamard difference families and skew Hadamard matrices

TL;DR

This paper generalizes classical constructions of skew Hadamard difference families, removing previous restrictions on parameters, and produces new infinite families of skew Hadamard matrices with broader applicability.

Contribution

It introduces a more general construction of skew Hadamard difference families that eliminates dependence on the exponent parameter, expanding the known classes of such matrices.

Findings

Existence of skew Hadamard difference families with 2^{u-1} blocks for certain prime powers.

Construction of new infinite families of skew Hadamard matrices.

Removal of previous parameter restrictions in classical constructions.

Abstract

In this paper, we generalize classical constructions of skew Hadamard difference families with two or four blocks in the additive groups of finite fields given by Szekeres (1969, 1971), Whiteman (1971) and Wallis-Whiteman (1972). In particular, we show that there exists a skew Hadamard difference family with $2^{u-1}$ blocks in the additive group of the finite field of order $q^e$ for any prime power $q\equiv 2^u+1\,({\mathrm{mod\, \, }2^{u+1}})$ with $u\ge 2$ and any positive integer $e$. In the aforementioned work of Szekeres, Whiteman, and Wallis-Whiteman, the constructions of skew Hadamard difference families with $2^{u-1}$ ($u=2$ or $3$) blocks in $({\mathbb F}_{q^e},+)$ depend on the exponent $e$, with $e\equiv 1,2,$ or $3\,({\mathrm{mod\, \, }4})$ when $u=2$, and $e\equiv 1\,({\mathrm{mod\, \, }2})$ when $u=3$, respectively. Our more general construction, in particular, removes the dependence on $e$. As a consequence, we obtain new infinite families of skew Hadamard matrices.